The apothem of a regular polygon is the distance from the center to any side.

Question

If the length of the apothem remains constant at 10 inches, the formula for the perimeter of a regular polygon as a function of the number of sides is ( ) ( )( ).
As the regular polygon changes from a pentagon (5 sides) to an octagon (8 sides), what is the approximate average rate of change in the perimeter?

Answer 1

If the perimeter is p, the area is A = 1/2 ap.

So, since p = ns, what is the side s of an n-gon?

The central angle of each of the n isosceles triangles is θ=360/n, so the side is

s = 20 tan(θ/2)

Thus, the area is

A = 1/2 a (ns) = 1/2 (10) (n*20 tan 180/n)
= 100n tan(180/n)

You can probably handle the rest, eh?

Answer 2

2+2=1